I can't say I know the details, but I can give you my thoughts which might give you a hint in the direction of an answer.
On a microscopic or even smaller level, materials are very rough. Two objects that touch, actually rarely actually do, and when they do, it's with very small and few contact points. You can imagine that the level of friction would not be ideal, however if you could fill those voids with a liquid of some kind, you could increase your traction between the surfaces. That in itself isn't the whole answer though, because lubrication to make things slip is doing exactly the same thing. So what's the difference?
In my opinion, that would be porosity. If we look at the surface of the paper in a microscope, it actually would look like a thick fiber or felt pad. Felt is something you place under furniture to make it slide more easily because it reduces pressure points. Also paper is perhaps slightly powdery I imagine on a microscopic level. Wetting your fingers will cause the moisture on your fingers to interact with all those fibers and slightly bind them together.
An other aspect is the surface texture of your fingers. Dry hands will slide more on smooth surfaces like glass compared to moist hands or sweaty hands, which I'm sure you could imagine would make a sticky squeaking sound. That's because your fingers are also porous and rough and fibery, not to mention while not exactly powdery, dry skin will more easily release dead skin cells which will further reduce traction. Dry skin is also more stiff and less rubbery. So wetting your skin will have the effect of softening the texture of your skin, making it rubbery, increase the surface contact with objects, reduce dead skin cells from getting in the way, bind loose skin fiber textures, and do the exact same thing to the paper.
An opposite effect is also true with paper. Handling non-porous materials like metal, your hands might get sweaty, increasing "sticktion" until there's so much sweat/moisture that it starts to lubricate instead and your hands slip. Porous materials like cloth, rope, wood, and paper, have the opposite effect by absorbing moisture and oils thus drying out the skin in contact, and so reducing traction even more than normal.
Hope that made sense, and sorry if it's a bit long.
I'm going with Surface Tension. Surface tension (or interfacial tension) being the function of a liquid to minimize it's surface (interface) energy. Solid-to-Water surface interfaces have surface tension just like an Air-Water interface. The key is that water experiences a lower surface tension with your finger pad or the page than with the air. The water molecules spread out between your finger and the page, minimizing the area exposed to air.
Surface tension is also responsible for things like capillary action, wicking, and enabling a drop of water to hold a coverslip to a microscope slide.